# Number of ways to choose two people out of 12 people

There are 12 people. We want to choose 2 of them so that their month of birth are different. What is the number of ways to do this? I think to solve the the problem, we need more information than given. Since we are choosing from unknown objects. If the problem was about the probability, then it would be simple. As we consider all of the possibilities of people's birth month.

• You are right, there is not enough information. They might just all have their birthdays in the same month, who knows? Maybe it got assumed that you have one person per month? Then again, that would make it rather easy to choose two with different months...
– Dirk
Apr 25 '17 at 10:59
• "If the problem was about the probability, then it would be simple." That's depepends on what the question is then. If you still want the number (as a stochastic variable) of ways to choose two people then it might not be that trivial anyway. Apr 25 '17 at 11:04

Assume that there are $m_i$ people born in month $i$, then the number of ways of choosing two born in different months is $$\frac 12\sum^{12}_1m_i(12-m_i)=72-\frac 12\sum^{12}_1m_i^2$$